A regular hexagon has a radius of 20 in. What is the approximate area of the hexagon?

Solution:

The diagram below is a regular hexagon inscribed in a circle. The radius of the circle is 20 inches. Hence OA = OB = 20 inches. 

The exterior angle of a regular polygon of n sides is 360°/n.

Since the hexagon is 6 sides the exterior angle of the hexagon is 360°/6 = 60°.

Hence ∠CBE = 60°. 

Therefore the interior angle ∠ABC = 180° - 60° = 120° = ∠BCD

Since it is a regular hexagon, we can assume that OB bisects the angle∠ABC, and hence ∠OBA = 120°/2 = 60°.

Since OB = OA (radius of the circle), we can conclude that ∠OAB = ∠OBA = 60°.

If two angles of the triangle △OAB are 60° then the third angle ∠AOB = 60°.

Since △OAB is an equilateral triangle (all angles equal) its three sides are also equal.

Hence AB = OA = OB = 20 inches

Now area of a triangle if all its three sides are given is given by the following expression:

Area of the triangle = \(\sqrt{s(s-a)(s-b)(s-c)}\)

Where s = semi-perimeter of the triangle = (a + b + c)/2

The semi-perimeter of the triangle OAB = (OA + OB + AB)/2 = (20 + 20 + 20)/2 = 30 inches

Therefore area of the triangle △OAB = \(\sqrt{30(30-20)(30-20)(30-20)}\)

= \(\sqrt{30(10)(10)(10)}\)

= 100√3 inches²

Since a regular hexagon has six triangles, as evident from the figure above, the total area of the hexagon inscribed inside the circle will be six times the area of △OAB

Area of the regular hexagon = 6 × 100√3 inches² = 600√3 inches² = 1039.2 inches²

A regular hexagon has a radius of 20 in. What is the approximate area of the hexagon?

Summary:

A regular hexagon has a radius of 20 in. The approximate area of the hexagon is 600√3 inches² = 1039.2 inches².